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We prove a general rigorous lower bound for$W(\Lambda,q)=\exp(S_0(\Lambda,q)/k_B)$, the exponent of the ground stateentropy of the $q$-state Potts antiferromagnet, on an arbitrary Archimedeanlattice $\Lambda$. We calculate large-$q$ series expansions for the exact$W_r(\Lambda,q)=q^{-1}W(\Lambda,q)$ and compare these with our lower bounds onthis function on the various Archimedean lattices. It is shown that the lowerbounds coincide with a number of terms in the large-$q$ expansions and henceserve not just as bounds but also as very good approximations to the respectiveexact functions $W_r(\Lambda,q)$ for large $q$ on the various lattices$\Lambda$. Plots of $W_r(\Lambda,q)$ are given, and the general dependence onlattice coordination number is noted. Lower bounds and series are alsopresented for the duals of Archimedean lattices. As part of the study, thechromatic number is determined for all Archimedean lattices and their duals.Finally, we report calculations of chromatic zeros for several lattices; theseprovide further support for our earlier conjecture that a sufficient conditionfor $W_r(\Lambda,q)$ to be analytic at $1/q=0$ is that $\Lambda$ is a regularlattice.Comment: 39 pages, Revtex, 9 encapsulated postscript figures, to appear in  Phys. Rev.

Lower bounds and series for the ground-state entropy of the Potts antiferromagnet on Archimedean lattices and their duals